What is a minus sign anyway?

Another my old post moved here from my old blog. I do that when I write something and need old stuff as a draft.

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A paper under this name is placed on the Internet by John Baldwin. He asks a seemingly naive question: how do we justify, in school mathematics teaching, manipulations like

(40+20) – (12+5) = (40-12) + (20-5)?

John Baldwin writes:

The associative law can only work on two applications of the plus sign. We generalize it to say we can regroup any sequences of additions. It is quite plausible that this is a distinction one should not make for teachers or at least the question should be at what level you want them to be aware of it. The second problem is that when minus signs are interspersed this gets more complicated. Since associativity fails for subtraction some further rules are required.

Indeed, notice that (3 -2) – 1 is not the same as 3 – (2-1). Subtraction is not associative!
This is a classical example of a “fuzziness” in mathematics teaching, a phenomen especially noticeable at earlier stages of school education. Very frequently, children are placed in position when they have to figure out rules which have not been made to them explicit. Of course, children learn the grammar of their mother tongue exactly that way, by absorption. Unfortunately, by the time they are taught mathematics, their natural ability to extract grammar rules from adult’s speech is already significantly suppressed.

Yes, of course, Benny Avelin is absolutely right. But what struck me in John Baldwin’s paper is his reluctance to bring this subtle destinction between the unary operation of negaation and the binary operation of subtraction to the attention of (American) teachers:

“It is quite plausible that this is a distinction one should not make for teachers or at least the question should be at what level you want them to be aware of it.”

In Britain, in a recently released Tender for Mathematics Professional Development Pilot, institutions are invited to submit proposals for a professional development course for teachers of mathematics who are not specialists in the subject teaching learners aged 11-19. Apparently (I picked this detail from a newspaper version of the advertisement), the course is to be run over 40 hours spread over 2 years. Will it give enough time to cover the subtraction/negation issue?

That was what struck me as well, this is why I wrote something that to me is the most straightforward way of thinking while maintaining mathematical correctness. This is something that all who have read an A course in math should know, which should imply that math teachers educated at the universities should know this.

But then again I have spoken to alot of teacher students, not even knowing what an operator is, and using dubious methods for calculating easy statements as these.

So the conclusion must be that the idea of this course for math teachers is a good one. Though I would say that the length of the course is far to short.

One of my great frustrations, at high school and in a university degree in pure mathematics, was the never-ending tendency of teachers of mathematics to pull rabbits from hats at the last moment. We are, they say, given all the axioms and inference rules needed to solve some problem or prove some conjecture, When we are unable to solve it or prove it, some additional result, method or “trick” is suddenly adduced.

This is one reason why a leading computer scientist, Edsgar Dijkstra, called pure mathematics a pre-industrial craft — still using ad-hoc methods when all other areas of human enquiry had long ago systematized their research procedures — and therefore unworthy of inclusion in a modern university.

I’d agree with the comments of Benny and Alexandre Borovik. However, what strikes me more is the complete lack of an attempt to derive the rules from the meaning (that is, counting of apples, say). A complete separation between semantics and syntax,
so to say, something which annoys people like Arnol’d.

A wonderful resource in early mathematical education (unfortunately, in Russian) is th ebook Maths for little ones by Alexandre Zvonkine. Zvonkine told me that he is considering a publication of an English translation of his book.

I stumbled across your comments about my notes on `what is a minus sign’. You are well to wonder when I question whether or maybe better how teachers should be taught
this. One of the faults of a preliminary version is that certain facts understood by the writer and immediate audience are not spelled out. Thus we were talking in the seminar for whom that was written about future elementary school teachers. And the issue that I was alluding to was at what stage you can make such students self-aware of subtle matters. One can’t just tell them that one operation is binary and the other unary.
Because they have never heard either word.

One of the members of the audience had been extremely successful in
teaching this group of students without making such formal distinction but by giving them lots of opportunity by operating on lattices of numbers to develop the understanding – analogously to, as you mentioned, children learning their native tongue.

In fact, I haven’t really resolved this issue in my own mind and recently have been working with high school teachers where the more explicit understanding is essential (and with some effort) communicable.